OFFSET
0,6
COMMENTS
The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.
LINKS
EXAMPLE
Triangle begins:
1
0 1
0 0 2
0 1 1 2
0 2 1 2 3
0 2 5 3 4 2
0 5 7 8 3 5 4
0 9 12 11 17 5 8 2
0 14 26 23 22 24 6 9 4
0 25 42 54 41 36 36 7 12 3
0 46 76 88 107 60 60 48 9 14 4
0 78 144 166 179 176 101 83 68 10 17 2
0 136 258 327 339 311 299 139 122 81 12 18 6
0 242 457 602 704 591 544 447 198 165 109 12 23 2
Row n = 6 counts the following compositions:
. (15) (24) (321) (42) (51) (6)
(141) (114) (312) (1122) (411) (33)
(132) (231) (1113) (11112) (3111) (222)
(123) (213) (2112) (2211) (111111)
(1212) (1311) (1221) (21111)
(1131) (12111)
(2121) (11211)
(11121)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], Total[First/@Split[#, UnsameQ]]==k&]], {n, 0, 15}, {k, 0, n}]
CROSSREFS
Column n = k is A000005, except a(0) = 1.
Row-sums are A011782.
Column k = 1 is A096569.
For length instead of sum we have A106356.
Other types of runs (instead of anti-):
- For leaders of identical runs we have A373949.
- For leaders of weakly increasing runs we have A374637.
- For leaders of strictly increasing runs we have A374700.
- For leaders of weakly decreasing runs we have A374748.
- For leaders of strictly decreasing runs we have A374766.
A003242 counts anti-run compositions.
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Aug 02 2024
STATUS
approved